Free Access
Issue
A&A
Volume 572, December 2014
Article Number A71
Number of page(s) 8
Section Catalogs and data
DOI https://doi.org/10.1051/0004-6361/201424273
Published online 28 November 2014

© ESO, 2014

1. Introduction

Other galaxies have become the most prominent battlefields in current astrophysical research, mainly due to the large and long-lasting photometric surveys. These surveys like MACHO or OGLE have discovered thousands of new eclipsing binaries in the Magellanic Clouds, hence, we know only about twice more eclipsing binaries in our own Milky Way than in other galaxies (see Pawlak et al. 2013; or Graczyk et al. 2011).

On the other hand, the chemical composition of the Magellanic Clouds differs from that of the solar neighborhood (e.g. Ribas 2004), and the study of the massive and metal-deficient stars in the Small Magellanic Cloud (SMC) checks our evolutionary models for these abundances. All eclipsing binaries analysed here have properties that make them important astrophysical laboratories for studying the structure and evolution of massive stars (Ribas 2004).

Eccentric eclipsing binaries (EEBs) with an apsidal motion can provide us with an important observational test of theoretical models of stellar structure and evolution. A long-term collection of the times of EEBs minima observed for several years throughout the apsidal motion cycle and a consecutive detailed analysis of the period variations of EEB can be performed, yielding both the orbital eccentricity and the period of rotation of the apsidal line with high accuracy (Giménez 1994). Many different sets of stellar evolution models have been published in recent years, such as for Maeder (1999), or Claret (2005); however, to distinguish between them and to test, which one is more suitable, it is still rather difficult. The internal structure constants, as derived from the apsidal motion analysis, could serve as one independent criterion. On the other hand, only stellar parameters for EEBs with an accuracy of 1% can be used to discriminate between the models.

Here, we analyse the observational data and rates of apsidal motion for eighteen SMC detached eclipsing systems. All these systems are early-type objects, having eccentric orbits, which also exhibits an apsidal motion. Similar studies of Large Magellanic Cloud (LMC) EEBs have been presented by Michalska & Pigulski (2005), by Michalska (2007), and recently also by Zasche & Wolf (2013). As far as we know, only several eclipsing binaries with apsidal motion were analysed in SMC galaxy until now: SC3 139376, SC5 311566 (Graczyk 2003), and nine other systems by North et al. (2010).

2. Observations of minimum light

Monitoring of faint EEBs in external galaxies became almost routine nowadays with quite moderate telescopes of 1–2 m class, which are equipped with a modern CCD camera. However, a large amount of observing time is needed, which is usually unavailable at larger telescopes. During the last two observational seasons, we have accumulated 2660 photometric observations and derived 29 precise times of minimum light for selected eccentric systems. New CCD photometry was obtained at the La Silla Observatory in Chile, where the 1.54-m Danish telescope (hereafter DK154) with the CCD camera and R filter was used (remotely from the Czech Republic).

All CCD measurements were reduced in a standard way using the bias frames and then the flat fields. The comparison star was chosen to be close to the variable one and with similar spectral type. A synthetic aperture photometry and astrometry software developed by Velen and Pravec Aphot, was routinely used for reducing the data. No correction for differential extinction was applied because of the proximity of the comparison stars to the variable and the resulting negligible differences in air mass and their similar spectral types.

The new times of primary and secondary minima and their respective errors were determined by the classical Kwee & van Woerden (1956) method or by our new approach (see Sect. 4.2). All new times of minima are given in Table A.1.

3. Photometry and light curve modelling

The core of our analysis lies on the huge photometric data sets, as obtained during the Macho (Faccioli et al. 2007), Ogle II (Wyrzykowski et al. 2004), and Ogle III (Graczyk et al. 2011) surveys. These photometric data were used both for minima time analysis and for light curve analysis. The method of how the individual times of minima for the particular system were computed is presented in Sect. 4.2. Our new observations obtained at the Danish 1.54-m telescope were used only for deriving the times of minima for the selected targets.

The analysis of the light curves (LC) for the systems was carried out using the program PHOEBE, ver. 0.31a (Prša & Zwitter 2005), which is based on the Wilson-Devinney algorithm (Wilson & Devinney 1971) and its later modifications, but some of the parameters have to be fixed during the fitting process. The albedo coefficients Ai remained fixed at value 1.0, the gravity darkening coefficients gi = 1.0. The limb darkening coefficients were interpolated from the van Hammes tables (van Hamme 1993), and the synchronicity parameters (Fi) were also kept fixed at values of Fi = 1. The temperature of the primary component was derived from the photometric indices or other sources (see below). The problematic issue of the mass ratio was solved by fixing q = 1 because no spectroscopy for most of these selected systems exists, and for detached eclipsing binaries the LC solution is almost insensitive to the photometric mass ratio (see e.g. Terrell & Wilson 2005).

4. Methods used for the analysis

4.1. Apsidal motion analysis

For the analysis, we used the approach as presented below.

  • 1.

    At the beginning, all of the available photometric data wereanalysed, resulting in a set of minima times. Preliminary apsidalmotion parameters were derived (with the assumptioni = 90°).

  • 2.

    Secondly, the eccentricity (e), argument of periastron (ω), and apsidal motion rate () that resulted from the apsidal motion analysis were used for the preliminary light curve analysis.

  • 3.

    As the third step, the inclination (i) from the LC analysis was used for the final apsidal motion analysis.

  • 4.

    Finally, the resulted e, ω, and values from the apsidal motion analysis were used for the final LC analysis.

Moreover, this simple approach was a bit complicated because the minima times were also derived using the light curve template (see the AFP method in Sect. 4.2). Hence, the LC solution from step 2 allows us to derive the better times of minima for the step 3. The whole process run iteratively until the changes are negligible (usually it was enough to run these four steps two times).

The O–C diagrams of all available times of minima were analysed using the method presented by Giménez & García-Pelayo (1983). This is a weighted least-squares iterative procedure, including terms in the eccentricity up to the fifth order. There are five independent variables determined in this procedure. The periastron position ω is given by the linear equation where is the rate of periastron advance, E is the epoch, and the position of periastron for the zero epoch T0 is denoted as ω0. The relation between the sidereal and the anomalistic period, Ps and Pa, is given by and the period of apsidal motion by .

All new precise CCD times of minima were used with a weight of 10 in our computation; some of our less precise measurements were weighted by a factor of five, while the poorly covered minima were given a weight of 1.

4.2. Method of minima fitting

thumbnail Fig. 1

How the AFP method works.

Open with DEXTER

We developed and routinely used a method for deriving the times of minima for selected stars observed during the MACHO and OGLE surveys. This semi-automatic fitting procedure (hereafter AFP) has harvested the fact that the number of data points obtained during these two photometric surveys is large (typically thousands of data points) but obtained during many orbital revolutions of the close pair (a so-called sparse photometry).

Therefore, we can construct the phased light curve of the eclipsing binary in different time epochs. If the apsidal motion is prominent in the system, the shape of the light curve also slightly varies between the different epochs.

The first step is to divide the whole data set of photometry into several different “subsets”, which are used for constructing the individual light curves. Then, we usually choose the data set closest to the half of the time interval covered with observations and use these data points for constructing the light curve to be analysed.

Then, this light curve is analysed using the PHOEBE code, and the theoretical light curve template is being constructed. This LC model is then being used for deriving the individual times of minima easily by fitting this phased light curve to the phased light curves for the individual data sets. The best fit is obtained with the simplex algorithm and the least squares fitting method by only shifting the theoretical and observed light curve in two axis (phase and magnitude). If the star has constant magnitude over the whole time range of our data, there is no need to fit the magnitude shift, and only one free parameter is computed. When we find the best fit, then the times of minima are computed easily according to the ephemerides for a particular data set. Of course, for eccentric orbit binaries, both primary and secondary minima are being computed separately.

For the input, there are the data points, the time intervals, the ephemerides, and also parameters of the method. These are the duration of eclipse (how large portion of the phase curve around minima is being used for computing), minimum number of data points (if lower, the minimum is not computed), and the depth of minima. If 1 / 5 of the depth of minima is covered with data points, then this particular minimum is being computed.

Hence, by using this technique, we can usually obtain both primary and secondary minima for each data subset from an original photometry file. Moreover, this method can also be used in these cases, where the minimum is covered only very poorly, or only a descent to the minimum is covered. In these cases, the classical Kwee-van Woerden method would not work properly, so we can obtain more useful data points. On the other hand, we would like to emphasize that the method is suitable only for systems with low eccentricity, where the shape of the light curve is changing only slightly. Otherwise, we have to construct a separate light curve template for each of the data subset.

The whole method is graphically shown in Fig. 1, where an illustrative example of OGLE-SMC-ECL-0720 is being presented. All of the derived times of minima are stored in Table A.1. There are also given the errors of individual minima times, which are being computed also by AFP in the following way. The set of different solutions was computed for a particular minimum with different parameters of the code (length of interval around each minimum used for the analysis, number of data points according to their precision, etc.), yielding a set of times of minima, which is usually more than 10. From these minima data set, an average and its variance were computed. The variance is then taken as an approximate error estimation for the particular minimum.

thumbnail Fig. 2

Light curves of the analysed systems, the data taken from the OGLE III survey, and the I filter.

Open with DEXTER

thumbnail Fig. 3

O–C diagram for the times of minima for the analysed systems. The continuous and dashed curves represent predictions for the primary and secondary eclipses, respectively. The individual primary and secondary minima are denoted by dots and open circles, respectively. Larger symbols correspond to the measurements, which were given higher weights.

Open with DEXTER

Table 1

Relevant information for the analysed systems.

Table 2

Light curve parameters for the analysed systems.

Table 3

Parameters of the apsidal motion for the individual systems.

5. Notes on individual systems

All of the eclipsing systems were analysed using a similar approach, hence we cannot focus on every star in detail. See Table 1 for information and cross-identification of these stars. The abbreviations of the star names were used for all of the systems for a better brevity. That is, OGLE-SMC-ECL-0720 was shortened as #0720, etc. Only the most important results are summarized below. The final light curve fits, and the O–C diagrams are presented in Figs. 2 and 3; the parameters are given in Tables 2 and 3. The whole set of eighteen analysed systems can be divided into a few subsets according to available spectral information.

The largest group in our sample of stars comprise these stars, which were never observed spectroscopically, hence no spectral classification or radial velocity study was published so far. These systems are #0781, #1001, #1298, #1407, #2225, #2251, #2524, and #5233. Most of them were discovered as eclipsing binaries by Udalski et al. (1998), Wyrzykowski et al. (2004), or Faccioli et al. (2007). Several of them were mentioned as eccentric ones with apsidal motion in some of the above mentioned papers. Owing to having no information about their spectra, we only roughly estimated the spectral types from the measurements in photometric filters, as seen in Table 1. These observations were usually taken from Massey (2002) and from the dereddened photometric indices the spectral types were estimated (Popper 1980; Ducati et al. 2001; or Cox 2000). For some of the systems, there resulted a non-negligible third light contribution (e.g. #2225, #2251).

For some of the systems, the spectral types were published, so we can use them for a better primary temperature estimation for a subsequent light curve analysis. These systems are #0720, #2534, #3677, #4955, #5422, and #5434 (to this group of stars, two systems #0888 and #3951 also belong, but these were given a special focus in the following subsections). These binaries were also discovered by Udalski et al. (1998) and Faccioli et al. (2007); for some of them, a short note about their apsidal motion was published. The spectral types for these systems given by Evans et al. (2004) and Bonanos et al. (2010) are in good agreement with our spectral types that are estimated from the dereddened photometric indices.

5.1. OGLE-SMC-ECL-2186

Two systems, #2186 and #3594, were even published with their light and radial velocity curves solutions. The first one (#2186) was analysed by Wyithe & Wilson (2001), who presented a preliminary light curve solution with an eccentric orbit with e = 0.068. Graczyk (2003) analysed the LC of #2186, yielding an eccentricity of 0.251, no third light, and the luminosity ratio of L2/L1 = 0.843. Wyrzykowski et al. (2004) presented a note about its apsidal motion but with no estimation of its period. Concerning the spectral type, Graczyk (2003) estimated the types of about O9V+O9V but dealt only with the photometry. Later, Hilditch et al. (2005) published its spectral type to be B0+B0-3 based on 15 spectra of the star. They also analysed the light curve, yielding a value of the eccentricity of the orbit to be 0.063. However, their LC solution is not very convincing due to poor fit of the secondary minimum.

5.2. OGLE-SMC-ECL-3594

The second system (#3594) was also studied by Wyithe & Wilson (2001), who included this star into their sample of SMC eclipsing binaries with the light curve solution, which result in orbital inclination of 88.9° and an eccentricity of 0.144. Hilditch et al. (2005) analysed the system in more detail, resulting in an orbital eccentricity of 0.19 (based on photometry and spectroscopy together) and the spectral types of both components as B1+B1-3.

5.3. OGLE-SMC-ECL-0888

The object #0888 was first mentioned by Wyrzykowski et al. (2004), who also noted about its apsidal motion. Its spectral type was derived to be about O9V by Evans et al. (2004). We found that the pure apsidal motion is not able to describe the O–C diagram in detail, hence another effect has also to be included. We also tried to fit the parabolic fit to the ephemerides, with the apsidal motion hypothesis (can be interpreted as a mass transfer between the components, despite improbable for detached binary). However, this fit was also not very satisfactory. Therefore, we used a different code that computes the apsidal motion parameters with the third-body orbit (a so-called “light travel time” effect), as seen in for example Irwin (1959) or Mayer (1990). Ten parameters were fitted (five from the apsidal motion, five from the third body hypothesis); thus, this approach led to an acceptable solution with the lowest sum of squares residuals. The final parameters of the fit are given in Tables 3 and 4; the complete O–C diagrams are shown in Figs. 3 and 4.

From the third-body parameters, we could also compute the mass function of the distant component, which resulted in f(m3) = 0.059 ± 0.015M. From this value, one can calculate a predicted minimal mass of the third body (i.e. assuming coplanar orbits i3 = 90°), which resulted in m3,min = 4.9M. If we propose such a body in the system, one can ask whether it is detectable somehow in the already obtained data. The period is long for continuous monitoring of the radial velocity changes, but detecting the third light in the light curve solution would be promising. Assuming a normal main sequence star, its luminosity would be of about only L3,min = 1−2% of the total system luminosity. Such a weak third light would be hardly detectable in our poor-quality photometric data, but it is worth of try. Hence, we performed a new light curve solution with a special focus on the value of the third light for a LC solution. The value was really obtained, and its value is not negligible at all. As one can see from the parameters presented in Table 2, the third light represents about one half of the total light. This finding naturally explains why both the eclipses are so shallow. On the other hand, one can ask to which body the estimated spectral type of O9V belongs. If the third body is the dominant source, this is probably the O9V component, but the primary temperature of 33 200 K was assumed using the O9V primary, which now seems to be incorrect. However, having no other relevant information about the individual spectral types, one cannot easily assume a different primary temperature. Thus, we can conclude that the third body is probably present and orbits around the EB pair on orbit which is mildly inclined from the originally assumed 90°. It is hard to say anything more about such a body because of the high errors of the parameters (period, third light, etc.). More precise photometry or radial velocities would be very welcome for a final confirmation of our hypothesis.

5.4. OGLE-SMC-ECL-3951

The object #3951 is a part of the SMC open cluster NGC 346. Its eclipsing nature and orbital period was first presented by Udalski et al. (1998). Later, the star was classified as B1V by Massey et al. (2012). From the period analysis, a weak quasi-periodic signal also on the residuals after subtraction of the apsidal motion hypothesis (see Fig. 5) resulted. However, the variation is still too spurious for any final confirmation yet and we have not even try to fit the data with any additional variation, as in the previous case.

thumbnail Fig. 4

O–C diagram of #0888 after subtraction of the apsidal motion term.

Open with DEXTER

thumbnail Fig. 5

O–C diagram of #3951 after subtraction of the apsidal motion term.

Open with DEXTER

Table 4

Third body orbit parameters for #0888.

6. Discussion and conclusions

Our study provides the parameters of the apsidal motion for eighteen early-type binary systems located in the SMC. For most of the binaries, this is the first attempt to estimate the apsidal motion rates, and the light curve solution. In our own Galaxy there are a few hundreds of apsidal motion eclipsing binaries known; however, in other galaxies their number is still very limited. Hence this study still presents an important contribution to the topic. However, for only three systems from our sample (#1001, #1298, and #5233), the apsidal motion was derived from adequately large data set covering almost one apsidal period. The relativistic effects for the selected systems are weak, being up to 10% of the total apsidal motion rate. For the system #0888, the third body hypothesis was also presented and discussed.

The apsidal motion in EEBs has been used for decades to test evolutionary stellar models. Thus, one can ask whether our results can be used for deriving the internal structure constants for these stars in SMC. However, dealing with no radial velocities for most of the systems and with rather poor data coverage during the apsidal motion period, the parameters are too uncertain and affected by large errors. For these systems where the apsidal period is well-covered, a detailed spectroscopic analysis is missing, and vice versa, for systems where the radial velocity study was performed, the apsidal period has yet to be only poorly covered with data. However, for any testing of the stellar structure models or for the general relativity tests, the quality of the input data has to be an order of magnitude better (which implies long-term collection of the observations and data that covers the whole apsidal period in the following decades). Some of the systems are bright enough for a spectral monitoring, hence we encourage the observers to obtain new, high-dispersion, and high-signal-to-noise spectroscopic observations. With such data, methods, like spectral disentangling, can help us construct the radial velocity curves of both components, confirm the apsidal motion hypothesis, test the stellar structure models, or detect the third bodies, as indicated from our analysis.

Acknowledgments

We do thank the MACHO and OGLE teams for making all of the observations easily public available. This work was supported by the Czech Science Foundation grant No. P209/10/0715, by the grant UNCE 12 of the Charles University in Prague, and by the grant LG12001 of the Ministry of Education of the Czech Republic. We are also grateful to the ESO team at the La Silla Observatory for their help in maintaining and operating the Danish telescope. The following internet-based resources were used in research for this paper: the SIMBAD database and the VizieR service operated at the CDS, Strasbourg, France, and the NASA’s Astrophysics Data System Bibliographic Services.

References

All Tables

Table 1

Relevant information for the analysed systems.

Table 2

Light curve parameters for the analysed systems.

Table 3

Parameters of the apsidal motion for the individual systems.

Table 4

Third body orbit parameters for #0888.

All Figures

thumbnail Fig. 1

How the AFP method works.

Open with DEXTER
In the text
thumbnail Fig. 2

Light curves of the analysed systems, the data taken from the OGLE III survey, and the I filter.

Open with DEXTER
In the text
thumbnail Fig. 3

O–C diagram for the times of minima for the analysed systems. The continuous and dashed curves represent predictions for the primary and secondary eclipses, respectively. The individual primary and secondary minima are denoted by dots and open circles, respectively. Larger symbols correspond to the measurements, which were given higher weights.

Open with DEXTER
In the text
thumbnail Fig. 4

O–C diagram of #0888 after subtraction of the apsidal motion term.

Open with DEXTER
In the text
thumbnail Fig. 5

O–C diagram of #3951 after subtraction of the apsidal motion term.

Open with DEXTER
In the text

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.